Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions

We compute the electromagnetic modes of a photonic crystal point defect by expansion in localized basis functions. The defect modes are expanded using Wannier functions formed from the eigenmodes of the surrounding photonic crystal. The underlying solution to the perfect crystal is based on a real-space solution to Maxwell's equations using the finite element method. The resulting method combines the efficiency and flexibility of using an unstructured grid to discretize the unit cell with the efficiency of the Wannier method, which provides a significant reduction in the number of unknowns (small bases), such as applied to the defect problems with strongly localized modes. We present numerical results for an example defect structure present in a crystal structure that offers convenient validation and comparison to other methods.